Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460301 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16361 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,12,6,13)(3,15,18,41,19,9,30,33)(4,20,24,10,25,37,23,36)(5,26,28,39,29,11,27,40)(16,44,21,31,46,34,49,42)(17,32,47,58,48,43,55,62)(22,50,53,35,54,60,52,38)(45,61,51,56,63,59,64,57);(1,3)(2,9)(4,21)(5,17)(6,18)(7,30)(8,19)(10,34)(11,32)(12,15)(13,33)(14,41)(16,24)(20,42)(22,51)(23,46)(25,49)(26,43)(27,47)(28,55)(29,48)(31,37)(35,59)(36,44)(38,61)(39,58)(40,62)(45,53)(50,57)(52,63)(54,64)(56,60);(1,4)(2,10)(3,16)(5,22)(6,24)(7,23)(8,25)(9,31)(11,38)(12,36)(13,37)(14,20)(15,42)(17,45)(18,49)(19,46)(21,30)(26,35)(27,53)(28,52)(29,54)(32,57)(33,44)(34,41)(39,50)(40,60)(43,56)(47,64)(48,63)(51,55)(58,59)(61,62);(1,5,8,29)(2,11)(3,17,19,48)(4,22,25,54)(6,27,7,28)(9,32)(10,35)(12,26)(13,39)(14,40)(15,43)(16,45,46,63)(18,47,30,55)(20,50)(21,51,49,64)(23,52,24,53)(31,56)(33,58)(34,59)(36,38)(37,60)(41,62)(42,57)(44,61)Polycyclic presentation:
2^9|a^2=g,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367498);